2020/01/15 by Burglind Jöricke, Jöricke, Burglind
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2001.05382
openalex publication_date 2020/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values λtr. The invariant descends to an invariant of elements of Bn\diagupZn, the braid group modulo its center. We prove that the number of elements of B3\diagupZ3 of positive λtr grows exponentially. The estimate applies to obtain effective finiteness theorems in the spirit of the geometric Shafarevich conjecture over Riemann surfaces of second kind. As a corollary we obtain another proof of the exponential growth of the number of conjugacy classes of B3\diagupZ3 with positive entropy not exceeding Y.